We describe a systematic framework for computing the conservative potential of a compact binary system using modern tools from scattering amplitudes and effective field theory. Our approach combines methods for integration and matching adapted from effective field theory, generalized unitarity, and the double-copy construction, which relates gravity integrands to simpler gauge-theory expressions. With these methods we derive the third post-Minkowskian correction to the conservative two-body Hamiltonian for spinless black holes. We describe in some detail various checks of our integration methods and the resulting Hamiltonian.

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Black Hole Binary Dynamics from the Double Copy and Effective Theory

Preface 1. Planck's radiation law and the Einstein coefficients 2. Quantum mechanics of the atom-radiation interaction 3. Classical theory of optical fluctuations and coherence 4. Quantization of the radiation field 5. Single-mode quantum optics 6. Multimode and continuous-mode quantum optics 7. Optical generation, attenuation and amplification 8. Resonance fluorescence and light scattering 9. Nonlinear quantum optics Index

The quantum theory of light.

Exact Solutions of Einstein's Field Equations: Notation

In Newtonian gravitational theory, a tidal Love number relates the mass multipole moment created by tidal forces on a spherical body to the applied tidal field. The Love number is dimensionless, and it encodes information about the body's internal structure. We present a relativistic theory of Love numbers, which applies to compact bodies with strong internal gravities; the theory extends and completes a recent work by Flanagan and Hinderer, which revealed that the tidal Love number of a neutron star can be measured by Earth-based gravitational-wave detectors. We consider a spherical body deformed by an external tidal field, and provide precise and meaningful definitions for electric-type and magnetic-type Love numbers; and these are computed for polytropic equations of state. The theory applies to black holes as well, and we find that the relativistic Love numbers of a nonrotating black hole are all zero.

http://absimage.aps.org/image/APR09/MWS_APR09-2009-000254.pdf

Relativistic theory of tidal Love numbers

Long ago, Newman and Janis showed that a complex deformation z → z + ia of the Schwarzschild solution produces the Kerr solution. The underlying explanation for this relationship has remained obscure. The complex deformation has an electromagnetic counterpart: by shifting the Coloumb potential, we obtain the EM field of a certain rotating charge distribution which we term $$ \sqrt{\mathrm{Kerr}} $$
 . In this note, we identify the origin of this shift as arising from the exponentiation of spin operators for the recently defined “minimally coupled” three-particle amplitudes of spinning particles coupled to gravity, in the large- spin limit. We demonstrate this by studying the impulse imparted to a test particle in the background of the heavy spinning particle. We first consider the electromagnetic case, where the impulse due to $$ \sqrt{\mathrm{Kerr}} $$
 is reproduced by a charged spinning particle; the shift of the Coloumb potential is matched to the exponentiated spin-factor appearing in the amplitude. The known impulse due to the Kerr black hole is then trivially derived from the gravitationally coupled spinning particle via the double copy.

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Kerr Black Holes as Elementary Particles

In this paper, we explore the physics of electromagnetically and gravitationally coupled massive higher spin states from the on-shell point of view. Starting with the three-point amplitude, we focus on the simplest amplitude characterized by matching to minimal coupling in the UV. In the IR, for charged states this leads to g = 2 for arbitrary spin, and the leading deformation corresponds to the anomalous magnetic dipole moment. We proceed to construct the (gravitational) Compton amplitude for generic spins via consistent factorization. We find that in gravitation couplings, the leading deformation leads to inconsistent factorization. This implies that for systems with Gauge2 = Gravity relations, such as perturbative string theory, all charged states must have g = 2. It is then natural to ask for generic spin, what is the theory that yields such minimal coupling. By matching to the one body effective action, we verify that for large spins the answer is Kerr black holes. This identification is then an on-shell avatar of the no- hair theorem. Finally using this identification as well as the newly constructed Compton amplitudes, we proceed to compute the spin-dependent pieces for the classical potential at 2PM order up to degree four in spin operator of either black holes.

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The simplest massive S-matrix: from minimal coupling to black holes

In this paper we compute the spin-dependent terms of the gravitational potential for general spinning bodies at the leading Newton’s constant G and to all orders in spin. We utilize the on-shell approach, which extracts the classical potential directly from the scattering amplitude. For spinning particles, extra care is required due to the fact that the spin space of each particle is independent. Once the appropriate matching procedures are applied, taking the classical-spin limit we obtain the potential for general spinning bodies. When the Wilson coefficients are set to unity, we successfully reproduced the potential for the Kerr black hole. Interestingly, for finite spins, we find that the finite-spin deviations from Kerr Wilson coefficients cancel with that in the matching procedure, reproducing the Kerr potential without the need for taking the classical-spin limit. Finally, we find that when cast into the chiral basis, the spin-dependence of minimal coupling exhibits factorization, allowing us to take the classical-spin limit straight forwardly.

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Classical potential for general spinning bodies

Building upon recent progress in applying on-shell amplitude techniques to classical observables in general relativity, we propose a closed-form formula for the conservative Hamiltonian of a spinning binary system at the 1st post-Minkowskian (1PM) order. It is applicable for general compact spinning bodies with arbitrary spin multipole moments. The formula is linear in gravitational constant by definition, but exact to all orders in momentum and spin expansions. At each spin order, our formula implies that the spin-dependence and momentum dependence factorize almost completely. We expand our formula in momentum and compare the terms with 1PM parts of the post-Newtonian computations in the literature. Up to canonical transformations, our results agree perfectly with all previous ones. We also compare our formula for black hole to that derived from a spinning test-body near a Kerr black hole via the effective one-body mapping, and find perfect agreement.

/pdf/complete-hamiltonian-for-spinning-binary-systems-at-first-4jngqr6l8c.pdf

Complete Hamiltonian for spinning binary systems at first post-Minkowskian order

Building upon recent progress in applying on-shell amplitude techniques to classical observables in general relativity, we propose a closed-form formula for the conservative Hamiltonian of a spinning binary system at the 1st post-Minkowskian (1PM) order. It is applicable for general spinning bodies with arbitrary spin multipole moments. The formula is linear in gravitational constant by definition, but exact to all orders in momentum and spin expansions. At each spin order, our formula implies that the spin-dependence and momentum dependence factorize almost completely. We expand our formula in momentum and compare the terms with 1PM parts of the post-Newtonian computations in the literature. Up to canonical transformations, our results agree perfectly with all previous ones. We also compare our formula for black hole to that derived from a spinning test-body near a Kerr black hole via the effective one-body mapping, and find perfect agreement.

/pdf/complete-hamiltonian-for-spinning-binary-systems-at-first-54g4qm3fgv.pdf

In this paper, we demonstrate that at leading order in post Minkowskian (PM) expansion, the stress-energy tensor of Kerr-Newman black hole can be recovered to all orders in spin from three sets of minimal coupling: the electric and gravitational minimal coupling for higher-spin particles, and the “minimal coupling” for massive spin-2 decay. These couplings are uniquely defined from kinematic consideration alone. This is shown by extracting the classical piece of the one-loop stress-energy tensor form factor, which we provide a basis that is valid to all orders in spin. The 1 PM stress tensor, and the metric in the harmonic gauge, is then recovered from the classical spin limit of the form factor.

/pdf/kerr-newman-stress-tensor-from-minimal-coupling-9pchs6jknr.pdf

Ming-Zhi Chung

Papers

The simplest massive S-matrix: from minimal coupling to black holes

Classical potential for general spinning bodies

Complete Hamiltonian for spinning binary systems at first post-Minkowskian order

Complete Hamiltonian for spinning binary systems at first post-Minkowskian order

Kerr-Newman stress-tensor from minimal coupling